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Faraday instability on a sphere: numerical simulation

Published online by Cambridge University Press:  10 May 2019

A. Ebo-Adou
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, 75005, France Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France Institut des Sciences de la Terre, Centre d’Études et de Recherche de Djibouti, Route de l’aéroport B.P 486 Djibouti-ville, République de Djibouti
L. S. Tuckerman*
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, 75005, France
S. Shin
Affiliation:
Department of Mechanical and System Design Engineering, Hongik University, Seoul 121-791, Republic of Korea
J. Chergui
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France
D. Juric
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France
*
Email address for correspondence: laurette@pmmh.espci.fr

Abstract

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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Ebo-Adou et al. supplementary movie 1

Visualisation of $\ell=1$ mode for spherical drop. The drop is displaced alternately to the left and the right. Length and colors of arrows indicate the velocity of the surrounding air.

Download Ebo-Adou et al. supplementary movie 1(Video)
Video 6 MB

Ebo-Adou et al. supplementary movie 2

Visualisation of $\ell=2$ prolate-oblate pattern of gravitational harmonic waves. Drop interface and velocity field on and outside drop are shown.

Download Ebo-Adou et al. supplementary movie 2(Video)
Video 49 MB

Ebo-Adou et al. supplementary movie 3

Visualisation of $\ell=3$ tetrahedral pattern of capillary subharmonic waves.

Download Ebo-Adou et al. supplementary movie 3(Video)
Video 2 MB

Ebo-Adou et al. supplementary movie 4a

Visualisation of $\ell=4$ cubic-octahedral pattern of capillary subharmonic waves.

Download Ebo-Adou et al. supplementary movie 4a(Video)
Video 1 MB

Ebo-Adou et al. supplementary movie 4b

Visualisation of $\ell=4$ axisymmetric pattern for subharmonic capillary waves. In this stroboscopic film, only snapshots at a single temporal phase are included, emphasizing the overall drift of the pattern

Download Ebo-Adou et al. supplementary movie 4b(Video)
Video 4 MB

Ebo-Adou et al. supplementary movie 5

Visualisation of $\ell=5$ mode for subharmonic capillary waves, showing evolution from axisymmetric to $D_4$ pattern.

Download Ebo-Adou et al. supplementary movie 5(Video)
Video 15 MB

Ebo-Adou et al. supplementary movie 6

Visualisation of $\ell=6$ icosahedral-dodecahedral pattern for subharmonic capillary waves.

Download Ebo-Adou et al. supplementary movie 6(Video)
Video 3 MB